Continuing on from the OP...
14 But continue thou in the things which thou hast learned and hast been assured of, knowing of whom thou hast learned them; 15 And that from a child thou hast known the holy scriptures, which are able to make thee wise unto salvation through faith which is in Christ Jesus. 16 All scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness: 17 That the man of God may be perfect, thoroughly furnished unto all good works. -- 2 Tim. 3:14-17 (KJV)
The Holy Scriptures which Timothy knew from a child was the Hebrew Bible (The Torah-Law, Prophets, and Writings). Paul says they are able to make someone wise until salvation through faith in the Messiah; they are inspired by God; profitable for doctrine, reproof, correction, and instruction in righteousness; so that a man of God may be perfect and thoroughly equipped to do all good works. He said, "ALL SCRIPTURE." The NT writings, while they are Scripture and we recognize them as such, were largely unwritten at this time, and they were canonized later. At a minimum, "ALL SCRIPTURE" must include the Hebrew Bible, which was the common baseline of accepted, compiled Scripture at the time. At a maximum, it includes that which was currently written, later written, and later canonized. The Hebrew Bible being the minimum, we should accept that this includes the commands of the Torah-Law. I believe it is completely divisive of the passage for one to claim that VV.14-15 is talking about the Hebrew Bible, and VV.16-17 is talking only about the NT writings. No, the Hebrew Bible is all-encompassing to this passage. This is practically the only way that Timothy would have understood it in the absence of much of the writing and the canonization of the NT writings. Yes, Peter calls Paul's writing Scriptures, and maybe Timothy could have thought those were included as well, but the baseline here is the Hebrew Bible. There is no way that Timothy would have thought "ALL SCRIPTURE" to mean only the very limited Scriptures at the time, to which he unlikely had access to study. And Paul clearly defines the Holy Scriptures as that which he had access to growing up. That includes the commands of the Torah-Law. These commands offer profitable instruction. Paul is saying study them, learn them, correct yourself by them, and then do good works as a result of your equipping! (And also, this does not preclude the example of the Messiah and guiding of the Holy Spirit, but the focus here is the Hebrew Bible).
The whole letter actually flows quite nicely: Paul first gives the instruction to hold fast to sound words, he later identifies the profitability of the Hebrew Bible, and finally warns against straying from sound doctrine. The Hebrew Bible certainly contains sound words, is profitable, and is sound doctrine. Within this context, who would Timothy have understood as having "itchy ears?"